Create a class to supersede operands_and_partials for N-ary operations

[SteveBronder]

Summary

This creates a class propagate_partials that is a drop in replacement for operands_and_partials. A new class was created so that stan math can stay backwards compatible. Underneath the hood propagate_partials is very similar to operands_and_partials, but now each edge is stored inside of a tuple. This has a small benefit of saving some memory relative to operands_and_partials since we previously allocated edges for 5 operations even if we only needed one. The main benefit is that this can support any number of operations and is a drop in replacement for operands_and_partials.

This change also makes for nicer code, where previously we had to include a lengthy template for operands_and_partials we can now just use auto and call the make_partials_propagator() function which then creates and returns the appropriate partials_propagator class. This is similar to the standard libraries make_tuple() function.

  // before
  operands_and_partials<decltype(y_col), decltype(mu_col), decltype(kappa_col)>
      ops_partials(y_col, mu_col, kappa_col);

  // now
  auto ops_partials = make_partials_propagator(y_col, mu_col, kappa_col);

Instead of operands_and_partials's edge1_ and edge2_ the partials_propagator class has an associated function called edge() which is used in the same way as std::tuple's std::get() function

// before
ops_partials.edge1_.partials_ = elt_divide(-P, Pi_cl);

// now
edge<0>(ops_partials).partials_ = elt_divide(-P, Pi_cl);

Tests

Since this is a drop in replacement for operands_and_partials I took all of the tests for that classe and wrote the corresponding ones for partials_propagator. They can be run via

./runTests.py -j2 ./test/unit/math/ -f partials_propagator

Side Effects

I changed all of the distributions to use make_partial_propagator so a lot of files are changed. I did this because I wanted to make sure everything worked and the easiest way was to swap them out for one another.

imo the files that really need looked at thoroughly are the operands_and_partials and partial_propagator ones in prim/fwd/rev. I'll double check the distributions match correctly.

Release notes

Allow for for N-ary partials for distributions

Checklist

• Math issue Adding 7-Parameter Drift Diffusion Model (DDM) PDF with partial derivatives to Stan Math #2682

• Copyright holder: Steve Bronder

  The copyright holder is typically you or your assignee, such as a university or company. By submitting this pull request, the copyright holder is agreeing to the license the submitted work under the following licenses:
  - Code: BSD 3-clause (https://opensource.org/licenses/BSD-3-Clause)
  - Documentation: CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

• the basic tests are passing

  • unit tests pass (to run, use: ./runTests.py test/unit)
  • header checks pass, (make test-headers)
  • dependencies checks pass, (make test-math-dependencies)
  • docs build, (make doxygen)
  • code passes the built in C++ standards checks (make cpplint)

• the code is written in idiomatic C++ and changes are documented in the doxygen

• the new changes are tested

[SteveBronder]

Staring at the code for using edge() and wondering if there should just be a function for directly getting the partials? So instead of

edge<0>(ops_partials).partials_ = elt_divide(-P, Pi_cl);

It's possible to just call

partials<0>(ops_partials) = elt_divide(-P, Pi_cl);

[SteveBronder]

@andrjohns would you mind taking a look at this? I've looked over all of the _lpdf/_lpmf files so I think if you look over the actual partial_propagator files and operands_and_partials files that would be good.

Not super high priority. This is mostly a nice cleanup. There may be some small optimizations for cases with only 1 or 2 parameters since we only store as many edges as we need now (and then only do reverse passes for those 1 or 2 parameters as well). Though I think we would only see a noticeable speedup when applying the scalar form of the lpdf many times.

[andrjohns]

You had me scared with the size of the diff, but overall the changes are pretty clean and minor! Just a few comments/suggestions

[andrjohns]

All LGTM! Feel free to merge if you don't have any other changes